The modular geometry of Random Regge Triangulations

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio

Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds

We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.Comment: 31 pages, 3 figures, partially replaces and extends arXiv:1010.099

Geometric Measure Theory with Applications to Shape Optimization Problems

This thesis mainly focuses on geometric measure theory with applications to some shape optimization problems being considered over rough sets. We extend previous theory of traces for rough vector fields over rough domains and proved the compactness of uniform domains without uniformly bounded perimeter assumption. As application of these results, together with some other tools from geometric analysis, we can give partial results on the existence and uniqueness of minimizers of the nematic liquid droplets problem and the thermal insulation problem. Two other geometric minimization problems with averaged property, which include the generalized Cheeger set problem, are also studied

Cohomology of Line Bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute equivariant cohomology for line bundles, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. This paper is considered the third in the row of arXiv:1003.5217 and arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Sasaki-Einstein 5-manifolds associated to toric 3-Sasaki manifolds

We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki moment map. It follows that there are infinitely many examples of these toric Sasaki-Einstein manifolds M for each odd b_2(M)>1. As an application of the proof of the above, we prove that the local deformation space of ASD structures on a compact toric ASD Einstein orbifold is given by Joyce ansatz conformal metrics.Comment: final corrections mad